Groups – Complexity – Cryptology

Abstract

The field of non-commutative group based cryptography has flourished in the past twelve years with the increasing need for secure public key cryptographic protocols. This has led to an active line of research called non-abelian group based cryptography. In this work, I in collaboration with Delaram Kahrobaei and Vladimir Shpilrain introduce a new public key exchange protocol based on a group theoretic problem and propose an appropriate platform group for this protocol. This work can be found in [19] and [20]. In addition, I in collaboration with Delaram Kahrobaei and Vladimir Shpilrain propose two new secret sharing schemes that utilize non-abelian groups. These schemes have some advantages over Shamir's secret sharing scheme (see [21] for the full paper). We propose a class of groups, namely small cancellation groups, to implement these secret sharing schemes. Choosing the platform groups used in group based cryptographic protocols is vital to their security. D. Kahrobaei and B. Eick proposed in [12] polycyclic groups as a potential platform for these cryptographic protocols. Polycyclic groups were also proposed as platform groups for group based cryptographic protocols in [28] and [10]. An important feature of polycyclic groups, and hence finitely generated nilpotent groups, is that they are linear. I in collaboration with Delaram Kahrobaei considered the complexity of an embedding of a finitely generated torsion free nilpotent group into a linear group (see [18]). We determined the complexity of an algorithm introduced by W. Nickel in [37] that determined a Q -basis for a finite dimensional faithful G-module, which gives a bound on the dimension of the matrices produced. In [18] we also modified Nickel's algorithm for building a Q -basis in order to improve the running time of the algorithm.